Fermat prime
If is a nonnegative integer the
Fermat number is defined to be
.
If is prime, then it is known as a Fermat prime. The first
Fermat numbers (for
) are known to be prime. Indeed:
Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime. However, this fails for
:
. In fact the primes listed above are the only Fermat numbers known to be prime.
Fermat primes are also the only primes in the form . This is easy to see, if
has an odd factor
then
, a contradiction, hence
is a power of
.